Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | C2×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
185 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,5)(6,7)(9,10,12,15)(11,13,16,14);(1,2,4,5)(3,7,8,6);(10,13)(11,16)(14,15);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^7|a^2=g,b^2=e,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=de,f^c=fg35 conjugacy classes — view the full character table.
| Fitting subgroup of | #3359⋊C3, #11712276, #11712281, … |
| maximal subgroup of | #19764, #19769, #19795, … |
| maximal normal subgroup of | #19764, #19769, #19795, … |
| maximal quotient of | Q8×#102, #22542, #22738, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2091);